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NAG Toolbox: nag_linsys_complex_posdef_tridiag_solve (f04cg)

Purpose

nag_linsys_complex_posdef_tridiag_solve (f04cg) computes the solution to a complex system of linear equations AX = BAX=B, where AA is an nn by nn Hermitian positive definite tridiagonal matrix and XX and BB are nn by rr matrices. An estimate of the condition number of AA and an error bound for the computed solution are also returned.

Syntax

[d, e, b, rcond, errbnd, ifail] = f04cg(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[d, e, b, rcond, errbnd, ifail] = nag_linsys_complex_posdef_tridiag_solve(d, e, b, 'n', n, 'nrhs_p', nrhs_p)

Description

AA is factorized as A = LDLHA=LDLH, where LL is a unit lower bidiagonal matrix and DD is a real diagonal matrix, and the factored form of AA is then used to solve the system of equations.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
Must contain the nn diagonal elements of the tridiagonal matrix AA.
2:     e( : :) – complex array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
Must contain the (n1)(n-1) subdiagonal elements of the tridiagonal matrix AA.
3:     b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr matrix of right-hand sides BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
The number of linear equations nn, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides rr, i.e., the number of columns of the matrix BB.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldb

Output Parameters

1:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
If ifail = 0ifail=0 or n + 1n+1, d stores the nn diagonal elements of the diagonal matrix DD from the LDLHLDLH factorization of AA.
2:     e( : :) – complex array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
If ifail = 0ifail=0 or n + 1n+1, e stores the (n1)(n-1) subdiagonal elements of the unit lower bidiagonal matrix LL from the LDLHLDLH factorization of AA. (e can also be regarded as the conjugate of the superdiagonal of the unit upper bidiagonal factor UU from the UHDUUHDU factorization of AA.)
3:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,n)ldbmax(1,n).
If ifail = 0ifail=0 or n + 1n+1, the nn by rr solution matrix XX.
4:     rcond – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the reciprocal of the condition number of the matrix AA, computed as rcond = 1 / (A1,A11)rcond=1/(A1,A-11).
5:     errbnd – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the forward error bound for a computed solution x^, such that x1 / x1errbndx^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and xx is the corresponding column of the exact solution XX. If rcond is less than machine precision, then errbnd is returned as unity.
6:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail < 0andifail999ifail<0andifail-999
If ifail = iifail=-i, the iith argument had an illegal value.
  ifail = 999ifail=-999
Allocation of memory failed. The double allocatable memory required is n. In this case the factorization and the solution XX have been computed, but rcond and errbnd have not been computed.
  ifail > 0andifailNifail>0andifailN
If ifail = iifail=i, the leading minor of order ii of AA is not positive definite. The factorization could not be completed, and the solution has not been computed.
W ifail = N + 1ifail=N+1
rcond is less than machine precision, so that the matrix AA is numerically singular. A solution to the equations AX = BAX=B has nevertheless been computed.

Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
(A + E) = b,
(A+E) x^=b,
where
E1 = O(ε) A1
E1=O(ε) A1
and εε is the machine precision. An approximate error bound for the computed solution is given by
(x1)/(x1) κ(A) (E1)/(A1) ,
x^-x1 x1 κ(A) E1 A1 ,
where κ(A) = A11A1κ(A)=A-11A1, the condition number of AA with respect to the solution of the linear equations. nag_linsys_complex_posdef_tridiag_solve (f04cg) uses the approximation E1 = εA1E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The total number of floating point operations required to solve the equations AX = BAX=B is proportional to nrnr. The condition number estimation requires O(n)O(n) floating point operations.
See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The real analogue of nag_linsys_complex_posdef_tridiag_solve (f04cg) is nag_linsys_real_posdef_tridiag_solve (f04bg).

Example

function nag_linsys_complex_posdef_tridiag_solve_example
d = [16;
     41;
     46;
     21];
e = [ 16 + 16i;
      18 - 9i;
      1 - 4i];
b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,  61 - 66i;
      78 - 80i,  71 - 74i;
      14 - 27i,  35 + 15i];
[dOut, eOut, bOut, rcond, errbnd, ifail] = nag_linsys_complex_posdef_tridiag_solve(d, e, b)
 

dOut =

    16
     9
     1
     4


eOut =

   1.0000 + 1.0000i
   2.0000 - 1.0000i
   1.0000 - 4.0000i


bOut =

   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i


rcond =

   1.0862e-04


errbnd =

   1.0221e-12


ifail =

                    0


function f04cg_example
d = [16;
     41;
     46;
     21];
e = [ 16 + 16i;
      18 - 9i;
      1 - 4i];
b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,  61 - 66i;
      78 - 80i,  71 - 74i;
      14 - 27i,  35 + 15i];
[dOut, eOut, bOut, rcond, errbnd, ifail] = f04cg(d, e, b)
 

dOut =

    16
     9
     1
     4


eOut =

   1.0000 + 1.0000i
   2.0000 - 1.0000i
   1.0000 - 4.0000i


bOut =

   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i


rcond =

   1.0862e-04


errbnd =

   1.0221e-12


ifail =

                    0



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Chapter Introduction
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